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Implications from propositions A => B, A <=> C and C => B

  1. Oct 6, 2007 #1
    I'm not 100% sure what this is in English so I'll try to describe it. Gives that:

    A: x^2 < 16
    B: -4 < x
    C: -4 < x < 4

    I'm supposed to put out every possibility for => and <=> between A,B and C. The key says that A => B, A <=> C and C => B. I can understand this, but isn't it true for every proposition (I think that it's called this) that A <=> A. That is, every proposition implies itself?
     
  2. jcsd
  3. Oct 6, 2007 #2

    CompuChip

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    Suppose you have three propositions, A, B and C and want to prove that they are equivalent. That means that
    A <=> B, B <=> C and A <=> C
    which can also be written as
    A => B, B => A, A => C, C => A, B => C, C => B.
    In words: if you know that one of them is true/false, they must all be true/false.
    Obviously, A <=> A is always true and it's not included in the list.

    You can prove all 6 of them consecutively, but that would be a lot of work. Therefore we find a shortcut:
    suppose you would be able to prove half of them, namely that A => B, B => C and C => A.
    Then because B implies C and C implies A, B also implies A so you automatically get B => A and therefore A <=> B.
    Similarly, C => A and A => B so C also implies B (via A) hence B <=> C.
    So proving these three statements will prove all six of them.

    This is what the key suggests. It is wrong though, in claiming that A => B, A <=> C and C => B suffices. For example, you cannot get B => A (there is no premise starting with B).
     
  4. Oct 13, 2007 #3

    HallsofIvy

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    I think you misunderstood the question. Nothing was said about A<=>B, B<=>C, C<=>A. there is nothing here that implies B=> A and the answer key does NOT suggest such a thing.
     
  5. Oct 14, 2007 #4

    CompuChip

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    Probably then I got confused by the formulation.
    What is meant by "put out every possbility"?
    And I don't think I see the relevance of the question in the first part of the post to the question "isn't it true for every proposition ...".

    So I apologize if I mislead you x-is-y, perhaps you can try to rephrase the question (or if someone else understands it, explain it to my numb mind)?
     
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